Morten Ørregaard Nielsen

The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange, stock, and bond markets

We study the forecasting of future realized volatility in the foreign exchange, stock, and bond markets from variables in our information set, including implied volatility backed out from option prices. Realized volatility is separated into its continuous and jump components, and the heterogeneous autoregressive (HAR) model is applied with implied volatility as an additional forecasting variable. A vector HAR (VecHAR) model for the resulting simultaneous system is introduced, controlling for possible endogeneity issues. We find that implied volatility contains incremental information about future volatility in all three markets, relative to past continuous and jump components, and it is an unbiased forecast in the foreign exchange and stock markets. Out-of-sample forecasting experiments confirm that implied volatility is important in forecasting future realized volatility components in all three markets. Perhaps surprisingly, the jump component is, to some extent, predictable, and options appear calibrated to incorporate information about future jumps in all three markets.

Finite Sample Comparison of Parametric, Semiparametric, and Wavelet Estimators of Fractional Integration

ABSTRACT In this paper we compare through Monte Carlo simulations the finite sample properties of estimators of the fractional differencing parameter, d. This involves frequency domain, time domain, and wavelet based approaches, and we consider both parametric and semiparametric estimation methods. The estimators are briefly introduced and compared, and the criteria adopted for measuring finite sample performance are bias and root mean squared error. Most importantly, the simulations reveal that (1) the frequency domain maximum likelihood procedure is superior to the time domain parametric methods, (2) all the estimators are fairly robust to conditionally heteroscedastic errors, (3) the local polynomial Whittle and bias-reduced log-periodogram regression estimators are shown to be more robust to short-run dynamics than other semiparametric (frequency domain and wavelet) estimators and in some cases even outperform the time domain parametric methods, and (4) without sufficient trimming of scales the wavelet-based estimators are heavily biased.

Likelihood Inference for a Nonstationary Fractional Autoregressive Model

This paper discusses model based inference in an autoregressive model for fractional processes based on the Gaussian likelihood. The model allows for the process to be fractional of order d or d b; where db > 1=2 are parameters to be estimated. We model the data X1;:::;XT given the initial values X 0 n; n = 0;1;:::, under the assumption that the errors are i.i.d. Gaussian. We consider the like- lihood and its derivatives as stochastic processes in the parameters, and prove that they converge in distribution when the errors are i.i.d. with suitable moment conditions and the initial values are bounded. We use this to prove existence and consistency of the local likelihood estimator, and to …nd the asymptotic distrib- ution of the estimators and the likelihood ratio test of the associated fractional unit root hypothesis, which contains the fractional Brownian motion of type II.

Estimation of fractional integration in the presence of data noise

A comparative study is presented regarding the performance of commonly used estimators of the fractional order of integration when data is contaminated by noise. In particular, measurement errors, additive outliers, temporary change outliers, and structural change outliers are addressed. It occurs that when the sample size is not too large, as is frequently the case for macroeconomic data, then non-persistent noise will generally bias the estimators of the memory parameter downwards. On the other hand, relatively more persistent noise like temporary change outliers and structural changes can have the opposite effect and thus bias the fractional parameter upwards. Surprisingly, with respect to the relative performance of the various estimators, the parametric conditional maximum likelihood estimator with modelling of the short run dynamics clearly outperforms the semiparametric estimators in the presence of noise that is not too persistent. However, when a non-zero mean is allowed for, it may reverse the conclusion.

Long Memory in Stock Market Volatility and the Volatility-in-Mean Effect: The FIEGARCH-M Model

We extend the fractionally integrated exponential GARCH (FIEGARCH) model for daily stock return data with long memory in return volatility of Bollerslev and Mikkelsen (1996) by introducing a possible volatility-in-mean effect. To avoid that the long memory property of volatility carries over to returns, we consider a filtered FIEGARCH-in-mean (FIEGARCH-M) effect in the return equation. The filtering of the volatility-in-mean component thus allows the co-existence of long memory in volatility and short memory in returns. We present an application to the daily CRSP value-weighted cum-dividend stock index return series from 1926 through 2006 which documents the empirical relevance of our model. The volatility-in-mean effect is significant, and the FIEGARCH-M model outperforms the original FIEGARCH model and alternative GARCH-type specifications according to standard criteria.

Local Whittle Analysis of Stationary Fractional Cointegration and the Implied–Realized Volatility Relation

I consider local Whittle analysis of a stationary fractionally cointegrated model. The local Whittle quasi maximum likelihood estimator is proposed to jointly estimate the integration orders of the regressors, the integration order of the errors, and the cointegration vector. The proposed estimator is semiparametric in the sense that it employs local assumptions on the joint spectral density matrix of the regressors and the errors near the zero frequency. I show that the estimator is consistent under weak regularity conditions, and, under an additional local orthogonality condition between the regressors and the cointegration errors, I show asymptotic normality. Indeed, the estimator is asymptotically normal for the entire stationary region of the integration orders, and, thus, for a wider range of integration orders than the narrow-band frequency domain least squares estimator of the cointegration vector, and it is superior to the latter estimator with respect to asymptotic variance. Monte Carlo evidence documenting the finite-sample feasibility of the new methodology is presented. In an application to financial volatility series, I examine the unbiasedness hypothesis in the implied–realized volatility relation.

Directional Congestion and regime switching in a long memory model for electricity prices

The functioning of electricity markets has experienced increasing complexity as a result of deregulation in recent years. Consequently this affects the multilateral price behaviour across regions with physical exchange of power. It has been documented elsewhere that features such as long memory and regime switching reflecting congestion and non-congestion periods are empirically relevant and hence are features that need to be taken into account when modeling price behavior. In the present paper we further elaborate on the co-existence of long memory and regime switches by focusing on the effect that the direction of possible congestion episodes has on the price dynamics. Under non-congestion prices are identical. The direction of possible congestion is identified by the region with excess demand of power through the sign of price differences and hence three different states can be considered: Non-congestion and congestion periods with excess demand in the one or the other region. Using data from the Nordic power exchange, Nord Pool, we find that the price dynamics and long memory features of the price series generally are rather different across the different states. Also, there is evidence of fractional cointegration at some grid points when conditioning on the states.

Optimal Residual-Based Tests for Fractional Cointegration and Exchange Rate Dynamics

A Lagrange multiplier test of the null hypothesis of cointegration in fractionally cointegrated models is proposed. The test statistic uses fully modified residuals to cancel the endogeneity and serial correlation biases, and standard asymptotic properties apply under the null and under local alternatives. With iid Gaussian errors, the asymptotic Gaussian power envelope of all (unbiased) tests is achieved by the one-sided (two-sided) test. The finite-sample properties are illustrated by a Monte Carlo study. In an application to the dynamics among exchange rates for seven major currencies against the U.S. dollar, mixed evidence of the existence of a cointegrating relation is found.

Efficient Likelihood Inference in Nonstationary Univariate Models

Recent literature shows that embedding fractionally integrated time series models with spectral poles at the long-run and/or seasonal frequencies in autoregressive frameworks leads to estimators and test statistics with non-standard limiting distributions that must be simulated on a case-by-case basis. However, we show that by embedding the models in a general I(d) framework the resulting estimators and tests regain all the desirable properties from standard statistical analysis. We derive the time domain maximum likelihood estimator and show that it is consistent, asymptotically normal, and under Gaussianity asymptotically efficient in the sense that it has asymptotic variance equal to the inverse of the Fisher information matrix. The three likelihood based test statistics (Wald, likelihood ratio, and Lagrange multiplier) are asymptotically equivalent and have the usual asymptotic chi-squared distribution and under the additional assumption of Gaussianity they are locally most powerful. In the special case where the dynamics of the model is characterized by a scalar parameter, we show that, in addition, the two-sided tests achieve the Gaussian power envelope of all invariant and unbiased tests, i.e. they are uniformly most powerful invariant unbiased. The finite sample properties of the tests are evaluated by Monte Carlo experiments. In contrast to what might be expected from the literature, the likelihood ratio test is found to outperform the Lagrange multiplier and Wald tests.

Nonparametric Cointegration Analysis of Fractional Systems with Unknown Integration Orders

In this paper a nonparametric variance ratio testing approach is proposed for determining the number of cointegrating relations in fractionally integrated systems. The test statistic is easily calculated without prior knowledge of the integration order of the data, the strength of the cointegrating relations, or the cointegration vector(s). The latter property makes it easier to implement than regression-based approaches, especially when examining relationships between several variables with possibly multiple cointegrating vectors. Since the test is nonparametric, it does not require the specification of a particular model and is invariant to short-run dynamics. Nor does it require the choice of any smoothing parameters that change the test statistic without being reflected in the asymptotic distribution. Furthermore, a consistent estimator of the cointegration space can be obtained from the procedure. The asymptotic distribution theory for the proposed test is non-standard but easily tabulated. Monte Carlo simulations demonstrate excellent finite sample properties, even rivaling those of well-specified parametric tests. The proposed methodology is applied to the term structure of interest rates, where, contrary to both fractional and integer-based parametric approaches, evidence in favor of the expectations hypothesis is found using the nonparametric approach.